In this paper, a diffusive prey-predator model with strong Allee effect growth rate and a protection zone $\Omega _0$ for the prey is investigated. We analyze the global existence, long time behaviors of positive solutions and the local stabilities of semi-trivial solutions. Moreover, the conditions of the occurrence and avoidance of overexploitation phenomenon are obtained. Furthermore, we demonstrate that the existence and stability of non-constant steady state solutions branching from constant semi-trivial solutions by using bifurcation theory. Our results show that the protection zone is effective when Allee threshold is small and the protection zone is large.
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[1] |
W. C. Allee, Principles of Animal Ecology, Saunders, RI, 1949.
![]() |
[2] |
H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent prediction: An abstraction that works, Ecology, 76 (1995), 995-1004.
![]() |
[3] |
R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551.
doi: 10.2307/1940007.![]() ![]() |
[4] |
R. G. Casten and C. G. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Diff. Equat., 27 (1978), 266-273.
doi: 10.1016/0022-0396(78)90033-5.![]() ![]() ![]() |
[5] |
R. H. Cui, J. P. Shi and B. Y. Wu, Strong Allee effect in a diffusive predator-prey system with a protection zone, J. Diff. Equat., 256 (2014), 108-129.
doi: 10.1016/j.jde.2013.08.015.![]() ![]() ![]() |
[6] |
Y. H. Du and X. Liang, A diffusive competition model with a protection zone, J. Diff. Equat., 244 (2008), 61-86.
doi: 10.1016/j.jde.2007.10.005.![]() ![]() ![]() |
[7] |
Y. H. Du, R. Peng and M. X. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Diff. Equat., 246 (2009), 3932-3956.
doi: 10.1016/j.jde.2008.11.007.![]() ![]() ![]() |
[8] |
Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Diff. Equat., 229 (2006), 63-91.
doi: 10.1016/j.jde.2006.01.013.![]() ![]() ![]() |
[9] |
S. B. Hsu, On global stability of a predator-prey system, Math. Biosci., 39 (1978), 1-10.
doi: 10.1016/0025-5564(78)90025-1.![]() ![]() ![]() |
[10] |
J. K. Hale,
Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.
doi: 10.1090/surv/025.![]() ![]() ![]() |
[11] |
C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Ent., 91 (1959), 385-398.
doi: 10.4039/Ent91385-7.![]() ![]() |
[12] |
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-New York, 1966.
![]() ![]() |
[13] |
P. E. Kloeden and C. Pötzsche, Dynamics of modified predator-prey models, Int. J. Bifurc. Chaos, 20 (2010), 2657-2669.
doi: 10.1142/S0218127410027271.![]() ![]() ![]() |
[14] |
Y. Lou, Some reaction diffusion models in spatial ecology, Sci. Sin. Math., 45 (2015), 1619-1634.
doi: 10.1360/N012015-00233.![]() ![]() |
[15] |
Y. Lou and B. Wang, Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment, J. Fixed Point Theory Appl., 19 (2017), 755-772.
doi: 10.1007/s11784-016-0372-2.![]() ![]() ![]() |
[16] |
Y. Li and M. X. Wang, Stationary pattern of a diffusive prey-predator model with trophic intersections of three levels, Nonlinear Anal. RWA, 14 (2013), 1806-1816.
doi: 10.1016/j.nonrwa.2012.11.012.![]() ![]() ![]() |
[17] |
R. M. May, Stability and Complexity in Model Ecosystems, Princeton Univ. Press, Princeton, 1973.
![]() |
[18] |
K. Mischaikow, H. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.
doi: 10.1090/S0002-9947-1995-1290727-7.![]() ![]() ![]() |
[19] |
N. Min and M. X. Wang, Qualitative analysis for a diffusive predator-prey model with a transmissible disease in the prey population, Comput. Math. Appl., 72 (2016), 1670-1689.
doi: 10.1016/j.camwa.2016.07.028.![]() ![]() ![]() |
[20] |
L. Nirenberg, Topics in Nonlinear Functional Analysis, Amer. Math. Soc., Providence, RI, 2001.
doi: 10.1090/cln/006.![]() ![]() ![]() |
[21] |
W.J. Ni and M. X. Wang, Dynamics and patterns of a diffusive Leslie-Gower prey-predator model with strong Allee effect in prey, J. Diff. Equat., 261 (2016), 4244-4272.
doi: 10.1016/j.jde.2016.06.022.![]() ![]() ![]() |
[22] |
W. J. Ni and M. X. Wang, Dynamicl properties of a Leslie-Gower prey-predator model with strong Allee effect in prey, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3409-3420.
doi: 10.3934/dcdsb.2017172.![]() ![]() ![]() |
[23] |
P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 919-942.
doi: 10.1017/S0308210500002742.![]() ![]() ![]() |
[24] |
F. Rao and Y. Kang, The complex dynamics of a diffusive prey-predator model with an Allee effect in prey, Ecol. Complex., 28 (2016), 123-144.
doi: 10.1016/j.ecocom.2016.07.001.![]() ![]() |
[25] |
M. X. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Phys. D, 196 (2004), 172-192.
doi: 10.1016/j.physd.2004.05.007.![]() ![]() ![]() |
[26] |
M. X. Wang, Nonlinear Elliptic Ppartial Differential Equations (in Chinese), Science Press, Beijing, 2010.
![]() |
[27] |
Y. X. Wang and W. T. Li, Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone, Nonlinear Anal. RWA, 14 (2013), 224-245.
doi: 10.1016/j.nonrwa.2012.06.001.![]() ![]() ![]() |
[28] |
J. F. Wang, J. P. Shi and J. J. Wei, Dynamics and pattern formation in a diffusive predator-prey systems with strong Allee effect in prey, J. Diff. Equat., 251 (2011), 1276-1304.
doi: 10.1016/j.jde.2011.03.004.![]() ![]() ![]() |
[29] |
Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu, The Introduction of Reaction-Diffusion Equations (in Chinese), Science Press, Beijing, 2011.
![]() |
[30] |
F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Equat., 246 (2009), 1944-1977.
doi: 10.1016/j.jde.2008.10.024.![]() ![]() ![]() |
[31] |
J. Zhou and C. L. Mu, Coexistence of a diffusive predator-prey model with Holling type-Ⅱ functional response and density dependent mortality, J. Math. Anal. Appl., 385 (2012), 913-927.
doi: 10.1016/j.jmaa.2011.07.027.![]() ![]() ![]() |